Reconstruction from Samples--The Math

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Let denote the th sample
of the original sound , where
is time in seconds. Thus, ranges
over the integers, and is the
sampling period in seconds. Thesampling
rate in Hertz (Hz) is just the reciprocal of the sampling
period,i.e.,

To avoid losing any information as a result of sampling, we must
assume is
band-limited to less than half the sampling rate. This means
there can be no energy in at
frequency or
above. We will prove this mathematically when we proveShannon’s
Sampling Theorem in §A.3
below.

Then we can say
is band-limited to less than half the sampling rate if and
only if for all . In
this case, Shannon’s sampling theorem gives us that can
be uniquely reconstructed from the samples
by summing up shifted, scaled, sinc
functions:

where

The sinc function is the
impulse response of the ideal lowpassfilter.
This means its Fourier transform is a rectangular window in the
frequency domain. The particular sinc function used here
corresponds to the ideal lowpass filter which cuts off at half the
sampling rate. In other words, it has a gain of 1 between
frequencies 0 and , and a gain of zero at all higher frequencies.

The reconstruction of a sound from its samples can thus be
interpreted as follows: convert the sample stream into a
weighted
impulsetrain, and pass that signal through an ideal lowpass
filter which cuts off at half the sampling rate. These are the
fundamental steps ofdigital to analog conversion (DAC). In
practice, neither the impulses nor the lowpass filter are ideal,
but they are usually close enough to ideal that you cannot hear any
difference. Practical lowpass-filter
design is discussed in the context ofband-limited interpolation.